The Capra-subdifferential of the ℓ₀ pseudonorm

The (Formula presented.) pseudonorm counts the nonzero coordinates of a vector. It is often used in optimization problems to enforce the sparsity of the solution. However, this function is nonconvex and noncontinuous, and optimization problems formulated with (Formula presented.)–be it in the objective function or in the constraints–are hard to solve in general. Recently, a new family of coupling functions–called Capra (constant along primal rays)–has proved to induce relevant generalized Fenchel-Moreau conjugacies to handle the (Formula presented.) pseudonorm. In particular, under a suitable choice of source norm on  (Formula presented.)–used in the definition of the Capra coupling–the function  (Formula presented.) is Capra-subdifferentiable, hence is Capra-convex. In this article, we give explicit formulations for the Capra-subdifferential of the  (Formula presented.) pseudonorm, when the source norm is a (Formula presented.) norm with (Formula presented.). We illustrate our results with graphical visualizations of the Capra-subdifferential of (Formula presented.) for the Euclidean source norm.